Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > madjusmdet | Structured version Visualization version GIF version |
Description: Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
Ref | Expression |
---|---|
madjusmdet.b | ⊢ 𝐵 = (Base‘𝐴) |
madjusmdet.a | ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) |
madjusmdet.d | ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) |
madjusmdet.k | ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) |
madjusmdet.t | ⊢ · = (.r‘𝑅) |
madjusmdet.z | ⊢ 𝑍 = (ℤRHom‘𝑅) |
madjusmdet.e | ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) |
madjusmdet.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
madjusmdet.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
madjusmdet.i | ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
madjusmdet.j | ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
madjusmdet.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
Ref | Expression |
---|---|
madjusmdet | ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | madjusmdet.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
2 | madjusmdet.a | . 2 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | |
3 | madjusmdet.d | . 2 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) | |
4 | madjusmdet.k | . 2 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) | |
5 | madjusmdet.t | . 2 ⊢ · = (.r‘𝑅) | |
6 | madjusmdet.z | . 2 ⊢ 𝑍 = (ℤRHom‘𝑅) | |
7 | madjusmdet.e | . 2 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) | |
8 | madjusmdet.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | madjusmdet.r | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | madjusmdet.i | . 2 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) | |
11 | madjusmdet.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) | |
12 | madjusmdet.m | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
13 | eqeq1 2827 | . . . 4 ⊢ (𝑘 = 𝑖 → (𝑘 = 1 ↔ 𝑖 = 1)) | |
14 | breq1 5071 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝐼 ↔ 𝑖 ≤ 𝐼)) | |
15 | oveq1 7165 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 − 1) = (𝑖 − 1)) | |
16 | id 22 | . . . . 5 ⊢ (𝑘 = 𝑖 → 𝑘 = 𝑖) | |
17 | 14, 15, 16 | ifbieq12d 4496 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) |
18 | 13, 17 | ifbieq2d 4494 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
19 | 18 | cbvmptv 5171 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
20 | breq1 5071 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁)) | |
21 | 20, 15, 16 | ifbieq12d 4496 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)) |
22 | 13, 21 | ifbieq2d 4494 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
23 | 22 | cbvmptv 5171 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
24 | eqeq1 2827 | . . . 4 ⊢ (𝑙 = 𝑗 → (𝑙 = 1 ↔ 𝑗 = 1)) | |
25 | breq1 5071 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝐽 ↔ 𝑗 ≤ 𝐽)) | |
26 | oveq1 7165 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 − 1) = (𝑗 − 1)) | |
27 | id 22 | . . . . 5 ⊢ (𝑙 = 𝑗 → 𝑙 = 𝑗) | |
28 | 25, 26, 27 | ifbieq12d 4496 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)) |
29 | 24, 28 | ifbieq2d 4494 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
30 | 29 | cbvmptv 5171 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
31 | breq1 5071 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝑁 ↔ 𝑗 ≤ 𝑁)) | |
32 | 31, 26, 27 | ifbieq12d 4496 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)) |
33 | 24, 32 | ifbieq2d 4494 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
34 | 33 | cbvmptv 5171 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 23, 30, 34 | madjusmdetlem4 31097 | 1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 1c1 10540 + caddc 10542 ≤ cle 10678 − cmin 10872 -cneg 10873 ℕcn 11640 ...cfz 12895 ↑cexp 13432 Basecbs 16485 .rcmulr 16568 CRingccrg 19300 ℤRHomczrh 20649 Mat cmat 21018 maDet cmdat 21195 maAdju cmadu 21243 subMat1csmat 31060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-splice 14114 df-reverse 14123 df-s2 14212 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-symg 18498 df-pmtr 18572 df-psgn 18621 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-rnghom 19469 df-drng 19506 df-subrg 19535 df-sra 19946 df-rgmod 19947 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-dsmm 20878 df-frlm 20893 df-mat 21019 df-marrep 21169 df-subma 21188 df-mdet 21196 df-madu 21245 df-minmar1 21246 df-smat 31061 |
This theorem is referenced by: mdetlap 31099 |
Copyright terms: Public domain | W3C validator |